# regCheck -- checking the regularity condition for a toric vector bundle

## Synopsis

• Usage:
b = regCheck E
• Inputs:
• Outputs:
• b, , whether E satisfies the regularity condition

## Description

"For a toric vector bundle in Kaneyama's description, the regularity condition means that for every pair of maximal cones $\sigma_1,\sigma_2$intersecting in a common codimension-one face, the two sets of degrees $d_1,d_2$ and the transition matrix $A_{1,2}$ fulfil the regularity condition. I.e. for every $i$ and $j$ we have that either the $(i,j)$ entry of the matrix $A_{1,2}$ is $0$ or the difference of the $i$-th degree vector of $d_1$ of $\sigma_1$ and the $j$-th degree vector of $d_2$ of $\sigma_2$ is in the dual cone of the intersection of $\sigma_1$ and $\sigma_2$."

Note that this is only necessary for toric vector bundles generated 'by hand' using addBaseChange and addDegrees, since bundles generated for example by tangentBundle satisfy the condition automatically.

 i1 : E = tangentBundle(pp1ProductFan 2,"Type" => "Kaneyama") o1 = {dimension of the variety => 2 } number of affine charts => 4 rank of the vector bundle => 2 o1 : ToricVectorBundleKaneyama i2 : regCheck E o2 = true

• addBaseChange -- changing the transition matrices of a toric vector bundle
• addDegrees -- changing the degrees of a toric vector bundle
• cocycleCheck -- checks if a toric vector bundle fulfills the cocycle condition
• isVectorBundle -- checks if the data does in fact define an equivariant toric vector bundle

## Ways to use regCheck :

• "regCheck(ToricVectorBundleKaneyama)"

## For the programmer

The object regCheck is .